Let $X$ be a topological space, and $U$ an open subset. Denote $j:U\to X$ the inclusion.
Let $\mathcal F$ be a sheaf on $U$. We define $j_!\mathcal F$ to be the sheaf associated to the presheaf $$\mathcal V\mapsto\left\{ \begin{array}{ll} \mathcal F(V) & \mbox{if } V\subseteq U \\ 0 & \mbox{otherwise} \end{array} \right.$$
I know that the stalk $(j_!\mathcal F)_x$ is equal to $\mathcal F_x$ for $x\in U$, and $0$ for $x\notin U$, and that $j_!\mathcal F$ is the only sheaf with this property.
Is it true that $j_!\mathcal F(V)=0$ whenever $V\nsubseteq U$? Is this true under suitable conditions on $X$?
I'm particularly interested in the case that $X=\mathbb P^1_\mathbb C$ and $\mathcal F=\mathcal O_X$.
No, it is not true. Here is a counterexample:
Take $X=\mathbb C$, $U=\mathbb D(0;1)$ (the open disk of radius $1$ centered at zero) and $\mathcal F=\mathcal C^\infty _U$, the sheaf of smooth functions on $U$ .
Let now $f\in \mathcal C^\infty (\mathbb C)$ be a bump function with compact support $supp(f)\subset U$, such that $f(0)=1$.
This function can be seen as a non-zero section $0\neq f\in \Gamma(\mathbb C,j_!(\mathcal C^\infty_U))$ and you have your counterexample with $V=\mathbb C\nsubseteq U$
Edit: a general result
Let $M$ be $\mathcal C^k$ manifold and $U\subset M$ a relatively compact open subset .
Then the restriction morphism $$\operatorname {res}:\Gamma(M, j_!(\mathcal C^k_U))\stackrel {\cong}{\to}\mathcal C^k_c(U):\phi\mapsto \phi|U$$ between the global sections of $j_!( \mathcal C^k_U)$ and the $\mathcal C^k$ functions defined only on $\text U$ and with compact support is an isomorphism .
This result explains the above example and may help build an intuition for the functor $j_!$.